The bethe-salpeter equation and unstable particles - Denmead Smith, J; Denmead Smith, John (1971)
﻿ Using certain assumptions about off-shell elements, we show that Green's functions have poles with factorizable residues at those values of the direct channel energy which produce resonance poles in the second-sheet of the S-matrix. An extension of the Wick-rotated Bethe-Salpeter equation (BSE) in the ladder approximation is obtained for elastic and complex energies, and an existence theorem is proved for an equation which is non-singular except on the threshold cut. This method does not require contour-distortions, and the continuation to complex energies is direct. We show also that the Bethe-Salpeter Heitler matrix has a factorization property, and derive the energy-analytic form of the BSE when a subtraction is required. An approximate 1-dimensional reduction of the BSE is described which gives more accurate results than previous methods. A theory of pi- and rho-mesons coupled via the Yang-Mills interaction is considered in the bootstrap limit, and this leads to an expression for the rho-field. An equivalent pion interaction can then be obtained which is similar to the chiral-invariant lagrangians for zero-mass pions, and it is used in a rough calculation. The bootstrap limit of the Green's function equations is also given. We give a new exact method of numerically inverting the BSE using the energy-analytic representation, and for the scalar equation our figures agree with Schwarz and Zemach's to within about 1 per cent. There is no additional difficulty with derivative couplings, and the procedure was used to calculate P-wave pi-pi scattering with a Bethe-Salpeter rho-exchange potential A self-consistent resonance in the direct channel was not predicted at any mass and cutoff.